How to make a lamp only half-cast shadows?
You can accomplish this by changing the material of the object casting the shadow and using the Light Path node:



Here, the principled shader would be the material of the object (replace with your material), and the factor of the mix shader determines how much of the shadow this object casts. (1.0 is no shadow, 0.0 is entire shadow).

If you're unable to change the material of the object, another approach would be to render the scene twice, once with shadows enabled for the light source and one disabled, then mix the results together. Obviously, this has the downside of needing to render the scene twice.